# Alternation of Rationals and Irrationals?

I'm in a lunch group at work of recreational math geeks and we came up with a question which we need help to resolve. I apologize in advance, if my explanation is not perfectly rigorous.

Given these two statements:

• Between any two rational number X and Y there is an irrational number
• Between any two irrational numbers X and Y there is a rational number
• Rational numbers are countably infinite and irrationals are uncountably infinite

Does that imply that the real numbers always alternate between the rationals and irrationals (i.e. rational -> irrational -> rational -> irrational...etc) no matter how close X and Y get?

It seems like the vastly larger number of irrationals implies that there are two irrationals that don't have a rational between them.

What can help us resolve this misunderstanding on our part?

• also between any two (ir)rational numbers is a(n) (ir)rational number Commented May 15, 2019 at 20:29
• @SteveED The set of real numbers is not countable... Rationals and irrationals don't "alternate" because you cannot have a numbered list of real numbers. You cannot take a given rational number and identity the next real number. What the results says is that the rationals are so well mixed within the set of reals that you cannot pick any interval, no matter how small, that does not have any rational number. Commented May 15, 2019 at 20:40
• One could well-order the real numbers, under ZFC. But I don't know how rationals fit into that picture. See math.stackexchange.com/questions/6501/…. Commented May 15, 2019 at 20:45